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Question #43851

Difficulty: 8/10 | 🟢 Free

The Question

A physical pendulum consists of a uniform solid metal disk of mass $M$ (in kg) and radius $R$ (in m). A massless rigid rod of length $L = R$ (in m) is attached to the edge of the disk such that it extends radially outward. The pendulum is pivoted at the free end of the rod and oscillates in a vertical plane with a small angular amplitude. The rod is made of INVAR, which has a negligible coefficient of thermal expansion ( $\alpha_{\text{invar}} \approx 0$ ), while the disk is made of a metal with a coefficient of linear thermal expansion $\alpha$ (in K $^{-1}$ ). If the temperature of the entire system is increased by a small amount $\Delta \theta$ (in K), what is the fractional change in the time period of oscillation, $\frac{\Delta T}{T}$ ? (Assume $\alpha \Delta \theta \ll 1$ )

Options

A.
$\frac{11}{36} \alpha \Delta \theta$
B.
$\frac{1}{2} \alpha \Delta \theta$
C.
$\frac{5}{9} \alpha \Delta \theta$
D.
$\frac{7}{36} \alpha \Delta \theta$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Physical Pendulum Inertia|Thermal Expansion of Area and Volume|Calculus (Moment of Inertia)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple thermal expansion with rotational dynamics, specifically avoiding the trap of assuming uniform geometric scaling.

A common mistake is to assume that since the time period of a simple pendulum scales as $T \propto \sqrt{L}$ , the fractional change is simply $\frac{1}{2}\alpha\Delta\theta$ .

However, because the pendulum is composite (an expanding disk on a non-expanding rod), the moment of inertia $I$ and the center of mass distance $h$ scale at different, non-standard rates.

Students must correctly apply the parallel axis theorem to the expanded geometry and use binomial approximations to differentiate the time period equation $T = 2\pi\sqrt{I/Mgh}$ , recognizing that $\Delta I/I \neq 2\Delta R/R$ in this specific non-uniform setup.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Physical Pendulum Inertia
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free